Definition of 2-CNF (a.k.a Krom) formula

Question

In my lecturer's notes, the following definition for a 2-CNF wff is given:

A 2-CNF formula, or Krom formula is a CNF formula F such that every clause has at most two literals.

However, there is no mention in the notes of how one might represent clauses containing a single literal L in the wff's corresponding implication graph (although I suppose one could add an edge from every other literal to L). Looking at Krom's original paper, the definition he seems to use is:

A 2-CNF formula, or Krom formula is a CNF formula F such that every clause has exactly two literals.

This definition would seem to make a lot more sense. Which definition is correct? Am I missing something?

Answer 1

A unit clause of the form $p$ is the same as the clause $p \lor p$, and so corresponds to the arrow $\lnot p \to p$.

As an example, consider the unsatisfiable CNF $$ p \land (\lnot p \lor q) \land \lnot q. $$ The implication graph contains the following edges: $$ \lnot p \to p \\ p \to q, \lnot q \to \lnot p \\ q \to \lnot q $$ These can be arranged in a cycle: $$ \lnot p \to p \to q \to \lnot q \to \lnot p $$ This cycle contains both $p$ and $\lnot p$ (and also both $q$ and $\lnot q$), showing that the formula is unsatisfiable.